Problem 18 - Calculus: Early Transcendental Functions

Question

Find the gradient field corresponding to \(f\) Use a CAS to graph it. $$f(x, y)=y \sin x$$

Step-by-Step Solution

Answer

The gradient field corresponding to \(f(x, y)=y \sin x\) is \((y \cos x, \sin x)\). A graphical representation can be produced using a Computer Algebra System (CAS).

Step 1: Compute Partial Derivatives

The gradient field is the vector field of the partial derivatives of \(f(x, y)\). Compute the partial derivative of \(f(x, y)\) with respect to \(x\) and then compute the partial derivative of \(f(x, y)\) with respect to \(y\). For \(f(x, y) = y \sin x\), these are given by: \[\frac{\partial f}{\partial x} = y \cos x\] and \[\frac{\partial f}{\partial y} = \sin x\].

Step 2: Form the Gradient Vector

Form the gradient vector from these partial derivatives. This is done by stacking the obtained partial derivatives into a vector. The gradient vector field of \(f(x, y)\) is represented as \[\nabla f = (\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y})\] Thus, the gradient field corresponding to \(f(x, y)\) is \[\nabla f = (y \cos x, \sin x)\].

Step 3: Graph the Gradient Field

The final step is to graph the gradient field. Plot the vector field \((y \cos x, \sin x)\) on a 2D coordinate system using a Computer Algebra System (CAS). Unfortunately, the text format is unable to include graphical images.

Key Concepts

Partial Derivatives

Partial derivatives are a way to measure how a function changes as one of its input variables change, while keeping the other variables constant.
For a function with multiple variables, such as a surface in 3D space, we can find the rate at which the function's value is changing along each dimension separately by using partial derivatives.
For instance, given a function like \( f(x, y) = y \sin x \), partial derivatives will help us understand how the function's value changes if we move slightly in the direction of the \( x \)-axis or \( y \)-axis, individually.

• Partial derivative with respect to \( x \): Measures the change in function due to a small change in \( x \).
• Partial derivative with respect to \( y \): Measures the change in function due to a small change in \( y \).

When we compute partial derivatives, it’s like taking the derivative of the function but treating all other variables as constants. This helps us to build the gradient needed for forming a gradient vector.

Vector Field

A vector field is a mathematical construct that assigns a vector to every point in a subset of space. It's like placing tiny arrows on a map: at each point, an arrow has a direction and magnitude. This concept helps in visualizing how vectors behave throughout a space.
For the given function \( f(x, y) = y \sin x \), we can form the gradient field, which itself is a type of vector field, using partial derivatives.
The gradient of \( f \), noted as \( abla f \), consists of two partial derivatives:

• \( \frac{\partial f}{\partial x} = y \cos x \)
• \( \frac{\partial f}{\partial y} = \sin x \)

The gradient field \( abla f = (y \cos x, \sin x) \) represents how steeply the function \( f \) increases or decreases around each point for both directions.

Computer Algebra System (CAS)

A Computer Algebra System (CAS) is a software tool that helps in performing symbolic mathematical computations.
It is very useful in solving complex mathematical problems quickly and accurately, without needing to do the calculations by hand.
In the context of our exercise, a CAS can be used to visualize the gradient field of a function like \( f(x, y) = y \sin x \). Using CAS, we can effortlessly produce a graphical representation of the vector field \( abla f = (y \cos x, \sin x) \) on a coordinate plane.
Some benefits of using a CAS include:

• Speeds up complex computations for large-scale data.
• Reduces the risk of human error.
• Explores and visualizes mathematical properties effectively.

In tasks involving gradient fields, CAS not only helps us calculate the gradients but also allows us to visualize the behavior of functions in space, facilitating better understanding of the field dynamics.